A coarse Gallai theorem

Marc Distel; Ugo Giocanti; Jędrzej Hodor; Clément Legrand-Duchesne; Piotr Micek

A coarse Gallai theorem

Combinatorics 2026-01-27 v1

Authors: Marc Distel , Ugo Giocanti , Jędrzej Hodor , Clément Legrand-Duchesne , Piotr Micek

Abstract

We prove that there exist functions $f$ and $g$ such that for all positive integers $k$ and $d$ , for every graph $G$ and every subset $A$ of the vertices of $G$ , either $G$ contains $k$ $A$ -paths such that vertices of different $A$ -paths are at distance at least $d$ in $G$ , or there exists a set $X$ of the vertices of $G$ with $|X|\leq f(k)$ such that every $A$ -path in $G$ contains a vertex of $B_G(X,g(k,d))$ .

Keywords

graph theory graph cycles function theory

Cite

@article{arxiv.2601.18439,
  title  = {A coarse Gallai theorem},
  author = {Marc Distel and Ugo Giocanti and Jędrzej Hodor and Clément Legrand-Duchesne and Piotr Micek},
  journal= {arXiv preprint arXiv:2601.18439},
  year   = {2026}
}

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